Problem 42 - Elementary Algebra

Question

For the following problems, solve each conditional equation. If the equation is not conditional, identify it as an identity or a contradiction. $$ 8 k+7=2 k+1 $$

Step-by-Step Solution

Answer

Question: Solve the equation $$8k + 7 = 2k + 1$$ and identify if it's a conditional equation, an identity, or a contradiction. Answer: The solution for k is $$k = -1$$. This is a conditional equation.

Step 1: Set up the equation

The given equation is $$8k + 7 = 2k + 1$$. Our goal is to solve for k.

Step 2: Move variables to the left side

Subtract 2k from both sides of the equation: $$8k - 2k + 7 = 2k - 2k + 1$$ Simplify: $$6k + 7 = 1$$

Step 3: Move constants to the right side

Subtract 7 from both sides of the equation: $$6k + 7 - 7 = 1 - 7$$ Simplify: $$6k = -6$$

Step 4: Solve for k

Divide both sides by 6: $$\frac{6k}{6} = \frac{-6}{6}$$ Simplify: $$k = -1$$ Since we obtained a specific value for k, this is a conditional equation and not an identity or a contradiction. The solution for k is $$k = -1$$.

Key Concepts

Algebraic Identities

When solving algebraic equations, understanding the concept of algebraic identities is crucial. An algebraic identity is an equation that is true for all values of the variables involved. For example, the identity \(a^2 - b^2 = (a+b)(a-b)\) holds true for any values of \(a\) and \(b\).

In the context of solving equations, if you simplify an equation and it becomes an identity, it means the original equation was true for every value of the variable, and thus, it doesn't have a specific solution. However, not all equations we encounter are identities; some are conditional, meaning they are only true for certain values, and our goal is often to find these specific values.

Contradiction in Algebra

In contrast to identities, a contradiction in algebra is an equation that has no solution because it is false for all possible values of the variable. An example of a contradictory equation is \(3 + 2 = 4\); no matter what operations you apply, this statement will never hold true.

Identifying a contradiction during the process of solving an equation tells you that there has been a mistake, or the equation itself is set up with a premise that can never be satisfied. Recognizing contradictions can save you from spending time trying to solve unsolvable problems.

Equation Solving Steps

Understanding the structured equation solving steps can greatly assist students in working through algebra problems. Here's a generalized approach:

• Identify and write down the given equation.
• Rearrange the equation to group like terms, often by moving all variable terms to one side and constants to the other.
• Simplify both sides of the equation as much as possible.
• Isolate the variable by undoing any addition, subtraction, multiplication, or division that has been applied to it.
• Solve for the variable and check your solution by plugging it back into the original equation.

Following these steps can help clarify the process and ensure you're systematically working toward the solution.